Students Use of Computational Thinking in Linear Algebrajrabin/publications/Computational...

Students’ Use of Computational Thinkingin Linear Algebra

Spencer Bagley1 & Jeffrey M. Rabin2

Published online: 25 November 2015# Springer International Publishing Switzerland 2015

Abstract In this work, we examine students’ ways of thinking when presented with anovel linear algebra problem. Our intent was to explore how students employ andcoordinate three modes of thinking, which we call computational, abstract, andgeometric, following similar frameworks proposed by Hillel (2000) and Sierpinska(2000). However, the undergraduate honors linear algebra students in our study usedthe computational mode of thinking in a surprising variety of productive and reflectiveways. This paper examines the solution strategies that the students employed to solvethe problem, emphasizing their use of the computational mode of thinking.

Keywords Justification . Linear algebra . Problem solving . Procedural understanding .

Conceptual understanding

Introduction

A first course in linear algebra plays a pivotal role in the mathematical education ofcollege students in STEM disciplines. It usually follows a calculus sequence that maybe predominantly computational in focus, and serves as a first encounter with manyelements of more advanced mathematics. These include the emphasis on precisedefinitions and formal proofs, the creation of an abstract axiomatic structure, the studyof objects that are not easy to visualize, multiple representations of those objects, andcomputational methods that require non-routine choices and interpretations of those

Int. J. Res. Undergrad. Math. Ed. (2016) 2:83–104DOI 10.1007/s40753-015-0022-x

* Spencer [email protected]

Jeffrey M. [email protected]

1 School of Mathematical Sciences, University of Northern Colorado, Campus Box 122,501 20th Street, Greeley, CO 80639, USA

2 Department of Mathematics, UC San Diego, 9500 Gilman Drive # 0112, La Jolla,CA 92093-0112, USA

http://crossmark.crossref.org/dialog/?doi=10.1007/s40753-015-0022-x&domain=pdf

representations. The kinds of flexible thinking required are foreign to many students,for whom “the fog rolls in” and the subject remains opaque (Carlson 1993).

Much of the literature on the learning of linear algebra documents students’ inabil-ities to solve basic problems, to move flexibly between the representations, to useabstract theorems in concrete situations, and even to speak or write the basic languageof linear algebra coherently. Despite much insight into the causes of their difficulties,and creative pedagogical suggestions, the overall impression remains pessimistic.

Based on our own teaching experience, and consistent with previous work (Hillel2000; Sierpinska 2000), we hypothesized that students must learn three major ways ofthinking, and how to coordinate them with each other, in order to succeed in linearalgebra. We term these abstract, geometric, and computational thinking. The studyreported here was designed to explore students’ use of these ways of thinking and theirability to coordinate them while solving a novel linear algebra problem. The initialresearch questions for the study were: What strategies do students use to solve a novellinear algebra problem (described below)? What are the uses, affordances, and con-straints of each mode of thinking? In what ways do students coordinate these modes ofthinking? As our analysis progressed, we noted a preponderance of computationalthinking, used by students in unexpectedly effective ways, and we were thus led to askin addition: In what productive ways did students use computational thinking? Thepresent paper concentrates on this question.

The students in our study were first year undergraduates enrolled in an Honors linearalgebra course at a large university in the southwestern United States. As such, they donot form a representative sample of the general population of linear algebra students,and we would not expect our results to generalize immediately to this larger population.However, our students successfully used computational thinking to solve the problemin creative ways and to justify their solutions. Our study shows what successful studentthinking in linear algebra can look like and provides a positive counterpoint to theliterature on student deficiencies in linear algebra.

Theoretical Background and Literature Review

Drawing upon our own teaching experience, we hypothesize that mastery of linearalgebra requires students to develop, and move flexibly between, three essential waysof thinking that we term abstract, geometric, and computational thinking. Our tripartitetaxonomy of modes of thinking is closely related to similar categories described byHillel (2000) and Sierpinska (2000). We now review their work and explain how ourown viewpoint builds upon it.

Hillel speaks of three modes of description of vectors and operators: the abstract,algebraic, and geometric modes. He defines the abstract mode as using the languageand concepts of the general formalized theory, including vector spaces, subspaces,linear span, dimension, operators, and kernels. His algebraic mode includes thelanguage and concepts of the more specific theory of Rn, including n-tuples,matrices, rank, solutions of systems of linear equations, and row space. Evenmore specifically, his geometric mode includes the language and concepts of 2-and 3-space, including directed line segments, points, lines, planes, andgeometric transformations.

84 Int. J. Res. Undergrad. Math. Ed. (2016) 2:83–104

Hillel (2000) traces many student difficulties to instructors’ propensity for “con-stantly shifting modes of description and notation” (p. 199) without alerting students tothis. He points out that choosing a basis, or changing basis, may require students to shiftmodes of description, and that over-reliance on a single mode can cause problems forstudents, for example when they inappropriately apply geometric intuition in dimen-sions higher than three. His three modes roughly correspond to ours, but refer tolanguages that can describe the basic objects of study, whereas ours refer directly tothe ways that students think; Hillel acknowledges that “it is possible, for example, forstudents to be working inRn (the algebraic mode of description) and to be using severalmodes of reasoning” (p. 195).

Sierpinska (2000), like us, discusses three modes of thinking or reasoning. We willshorten her names to (analytic-)structural, (synthetic-)geometric, and(analytic-)arithmetic. While she does not give explicit definitions of these categories,she provides a wealth of examples. Her categories roughly correspond to Hillel’sabstract, geometric, and algebraic modes, respectively. She says these modes are“equally useful, each in its own context, and especially when they are in interaction”(p. 233) and traces student difficulties to “their inability to move flexibly between thethree modes” (p. 209). Her structural mode corresponds to our abstract one, and hergeometric mode to ours. However, we intend our computational mode to be moreinclusive than her arithmetic mode. As we elaborate below, our computational modeincludes reasoning about computations, which Sierpinska views as overlapping withthe other modes of thinking. Such computational reasoning, as we call it, will be acentral theme of this paper.

Our Tripartite Taxonomy

We use a similar threefold classification of students’ modes of thinking, comprisingabstract, geometric, and computational thinking; we present examples of each in theResults section. Abstract thinking treats vectors as abstract objects manipulated ac-cording to formal rules, without reference to components or arrows. It makes use ofdefinitions and theorems stated in coordinate-free language and applicable to Rn forany n. (Our course did not cover abstract vector spaces.) It makes assertions ofexistence or uniqueness on the basis of general principles rather than resulting fromexplicit computations. The notion of orthogonality may be part of abstract thinking ifits meaning comes from an abstract inner product rather than the specifically geometricnotion of right angles.

Geometric thinking involves visualization in Euclidean two- or three-dimensional space, with vectors represented as arrows, and may draw onconcrete facts from high school geometry such as the Pythagorean Theorem.Span and linear independence can be geometric concepts if based on geometricproperties such as collinearity and coplanarity in R2 and R3. “Orthogonal” as asynonym for “perpendicular” is a geometric concept. Systems of linearequations can be viewed geometrically in terms of the incidence relationsamong the lines or planes obtained by graphing each equation. A desirableoutcome of a linear algebra course is for students to learn to extend theirgeometric thinking to apply in Rn for n>3; we do not consider “geometricthinking” in such contexts to be an oxymoron.

Int. J. Res. Undergrad. Math. Ed. (2016) 2:83–104 85

Since the present study emphasizes students’ use of computational thinking, wewish to focus on our definition of computational thinking and its relation toSierpinska’s (2000) arithmetic mode of thinking. Computational thinking representsvectors in Rn explicitly as n-tuples of real numbers, and draws conclusions fromalgorithms such as row reduction of matrices for solving systems or computingdeterminants. Assertions of existence or uniqueness come from explicit computationsproducing the objects in question. Systems of linear equations are thought of in terms oftheir coefficient or augmented matrices. Most importantly, for us computational think-ing is not simply the procedural knowledge of how to execute an algorithm withouterrors. It includes choosing the appropriate computation to answer a particular question,understanding what the result of the computation means in that context, and reasoningabout the steps or the outcome of the computation.

In essence, our computational mode of thinking is an expansion of Sierpinska’s(2000) arithmetic mode to include reasoning about computations. Although Sierpinskaincludes some reasoning and proof within her arithmetic mode of thinking, its preciselimits are not clear to us, and seem too restrictive. On the one hand, she says that “Muchof the analytic-arithmetic reasoning goes along the line: Show that two processes leadto the same result” (p. 234). On the other, she gives an example she considers to beintermediate between arithmetic and structural thinking, in which “the student was notperforming calculations, he was reflecting on the properties of the possible effects of acalculation” (p. 240). We definitely consider this within the scope of computationalthinking. Regarding a second example of student reasoning, about the algorithm forinverting a matrix, she says that it is partly arithmetic but has “one foot in the structuralmode because it reflects on the technique and makes use of its reversible character” (p.241). We would again consider this within the scope of computational thinking.

Problems in linear algebra may require an insight or reasoning process that is mostaccessible via one specific way of thinking. However, coordination between multipleways of thinking is often required. By this we mean the ability to flexibly move fromone way of thinking to another, or more specifically to “import” information acquiredvia one mode into another mode for further reasoning. Translation to the geometricmode often provides intuitive confirmation or understanding of abstract or computa-tional results; a geometric picture may provide the key idea for an abstract proof oridentify a key quantity to be computed. For instance, a geometric viewpoint on leastsquares and orthogonal projection is an invaluably illuminating complement to anabstract or computational one. The abstract mode can provide necessary or sufficientconditions for drawing some conclusion, which conditions can then be verified com-putationally. Our study was initially designed to explore students’ ability to coordinatemultiple modes of thinking in solving a novel problem, and the affordances providedby such coordination, which is a feature of “expert” thinking in linear algebra. Indelineating the “boundaries” between the modes of thinking, and studying theirinteraction at these boundaries, we consider it important that each mode be definedbroadly enough to include reasoning processes carried out within its own “language”.In particular, this is the reason for our expansion of Sierpinska’s (2000) arithmetic modeof thinking into our computational category.

We view the classification of student thinking in linear algebra as computational,geometric, or abstract as being largely orthogonal to the broader classification ofstudents’ knowledge as procedural or conceptual. Students thinking in any of the


modes may draw upon knowledge anywhere on the continuum from procedural toconceptual. In particular, although computational thinking would include superficialprocedural knowledge, in which students apply rote algorithms without reflection(Hiebert and Lefevre 1986), we do not view computational thinking as limited tosuperficial procedural knowledge. At its best, computational thinking can be anexample of what Baroody et al. (2007), responding to Star (2005), call deep proceduralknowledge. For Baroody et al. (2007), “(relatively) deep procedural knowledge cannotexist without (relatively) deep conceptual knowledge” (p. 123). The “proceduralflexibility and critical judgment” that for us can be exhibited by students usingcomputational thinking “require the integration of procedural knowledge with concep-tual knowledge” (Baroody et al., p. 121). We will present examples of our studentssuccessfully and flexibly applying computational thinking to solve the problem weposed in a variety of creative and reflective ways.

Literature Review

The early literature on linear algebra tends to emphasize identification and diagnosis ofstudent errors and deficiencies. Hillel (2000) and Sierpinska (2000) highlight students’inability to utilize and flexibly coordinate the three modes of thinking. Dorier andSierpinska (2001) also point out students’ unfamiliarity with axiomatic frameworks andthe need to think in terms of formal definitions and general properties. Maracci (2008)studied student work on a challenging problem about the intersection of subspaces of avector space of dimension at least 5. He suggested that their difficulties reflect aninsufficiently general “paradigmatic model” of subspace limited to the span of aselected subset of given basis vectors, like the coordinate subspaces xi=0 of Rn. Healso pointed out the need to view a linear combination both as a process and as anobject. Stewart and Thomas (2010) frame their work in terms of APOS theory andTall’s three worlds of mathematics (embodied, symbolic, and formal) which theycompare to Hillel’s three modes of description. They point out that students are oftennot given time and opportunities to develop links between the three worlds, and thattheir procedural knowledge is not deep in the above sense: “students who thought thatthey should row reduce a matrix often did not know why, or what to do withthe result” (p. 186).

More recent work presents a more optimistic picture of student understanding inlinear algebra. Several studies have explored sociocultural aspects of student learning,and proposed new instructional approaches to enrich students’ concept images in linearalgebra. Wawro (2014) documented the evolution of student argumentation, at the levelof the classroom community, concerning the Invertible Matrix Theorem over the courseof a semester. This provides insight as to how students use and modify theirunderstanding of key concepts like span and independence as they make connectionsbetween the many criteria for a matrix to be invertible. Wawro et al. (2012) describedthe implementation of an instructional sequence in the spirit of Realistic MathematicsEducation to build rich concept images for span and independence, in which vectors areidentified with modes of travel through space. They showed that students continue todraw on these concrete images later as they work on more abstract tasks. Possani et al.(2010) similarly used a realistic traffic modeling task to generate concept images forsystems of linear equations and their solution methods.


Student problem-solving has also been examined in the framework of Sierpinska’sthree modes of thinking. Dogan-Dunlap (2010) coded students’ different approaches todetermining whether given sets of vectors were linearly independent and categorizedthese within Sierpinska’s three modes of thinking. Students’ use of geometric thinkingwas of most interest inasmuch as they were provided with software generating visualrepresentations of the vectors in question. Celik (2015) similarly coded approaches to amore abstract task asking whether the (in)dependence of a set of three unspecifiedvectors implies the (in)dependence of the new set obtained by deleting one vector, or byadding another. This latter task has some relation to our task (see below) of extending agiven set of vectors to a basis. Both of these authors found that arithmetic(computational) thinking predominated, which is consistent with our finding here.

Materials and Methods

The students in our study were enrolled in an Honors Linear Algebra course in theirfirst year at a large university in the southwestern United States. The course forms thefirst quarter of a three-quarter Honors Calculus sequence in which linear algebraprovides the conceptual basis for treating multivariable calculus in any number ofdimensions. Completion of Advanced Placement (AP) calculus in high school, andobtaining the highest possible score of 5 on the AP calculus BC examination, areprerequisites for the course. The instructor was not an author of this paper, although wedid observe his class on a few occasions and asked him some questions about thecourse content. One of us has taught the course in previous years, including the year ofthe pilot study mentioned below. Both the textbook (Hubbard and Hubbard 2009) andthe instructor of this iteration of the course take a fairly computational approach tolinear algebra, building much of the subject around the key idea of finding solutions tolinear systems.

Eight students (of 34 enrolled) responded to our request for volunteers to participatein a clinical interview (Ginsburg 1997); all of these were accepted. The eight students’course grades ranged from A though C. Although all volunteers were male, this was notunrepresentative of the class, which included only a few female students. Students wereinterviewed individually; the interviews lasted about one hour and took place at the endof the course, when essentially all course material had been covered.

The interview revolved around the following task, termed the “Michelle Problem:”

Michelle would like to create a basis for R4. She has already listed two vectors vand w that she would like to include in her basis, and wants to add more vectorsto her list until she obtains a basis. What instructions would you give her on howto accomplish this?

We had included this problem as one among many interview questions in an earlierpilot project conducted near the end of a previous year’s iteration of the same honorslinear algebra course (Wawro et al. 2011). Based on the responses at that time, wedetermined that the task has the potential to elicit all three modes of thinking:computational, geometric, and abstract. Although some textbooks do prove that alinearly independent set of vectors can always be extended to a basis, this was not


covered in the course the students took, and thus the problem was novel to them. Theframing in R4 rather than R3 is intended to discourage an immediate appeal to visualgeometric intuition, although geometric thinking is still applicable. Not providingnumerical components for the “given” vectors may facilitate an abstract approach,and by asking for “instructions” for Michelle we hope that students will reflect on theirmethods and perhaps present them in algorithmic form. The semistructured interviewswere conducted by one of the authors, joined by another colleague on some occasions.

Most of the eight students did not make much progress on the problem in the verygeneral way in which it was initially presented, with generic vectors v and w. Whenthey appeared “stuck” we asked them to test, or continue to develop, their ideas usingthe specific vectors v=[1 2 3 4] and w=[0 -1 4 2]. We asked a number of follow-upquestions to probe students’ intuition and solution procedures, whether students couldformulate their solution procedure in an algorithmic way, and whether theycould justify their procedure. More details of the interview protocol are pro-vided in Appendix 1.

The interviews were videotaped and transcribed, and students’ written work pro-duced during the interviews was retained. These recordings, transcripts, and writtendocuments formed the corpus of data analyzed in this study. Using grounded theory(Strauss and Corbin 1994), we coded students’ utterances as instances of abstract,computational, or geometric thinking, referring to written work for confirmatoryevidence, and documented the ways in which students used each mode of thinking.To code students’ utterances, we attended to vocabulary indicative of specific modes;for example, we usually coded utterances including “plane” and “perpendicular” asinstances of geometric thinking, whereas we usually coded utterances including “pivot”and “row reduction” as instances of computational thinking. Some terms, especially“orthogonal”, are meaningful in any mode, and we used context to resolve suchambiguity. As an example of the use of context, two students in our pilot study usedthe term “perpendicular” in the context of taking the cross product of two vectors or ofthe Gram-Schmidt process; based on the context, we could not conclude that they werethinking geometrically, since it was just as likely that they were thinking about analgorithm. References to the components of vectors, or the entries of matrices, wereusually taken to indicate the computational mode. We coded statements of theorems orproperties in coordinate-free language as instances of abstract thinking. We alsoattended to notations, procedures, images, or metaphors students used. In particular,our students often explicitly carried out the steps of an algorithm such as row reduction,or explicitly reasoned about those steps; this was one of our primary indications thatthey were using the computational mode of thinking. Similarly, reasoning based onvisual images or Euclidean geometry provided evidence of geometric thinking.

Results

The students in our interviews were largely successful in solving this task; all but one ofthe eight students developed a solution procedure. The most frequently used solutionmethod was some variant on the guess-and-check method; however, students usuallydeveloped strategies for producing guesses that they thought were more likely tosucceed. A summary of solution methods and the number of students who used each


is presented in Appendix 2. Throughout this section, we will capitalize the name ofeach mode of thinking to more clearly mark them.

Most of our results document students using Computational thinking in ways wefound surprisingly effective. As a contrast to those results, and to exemplify our coding,we begin by documenting some productive ways students used Abstract and Geometricthinking. Most of the students began with the Abstract definition of basis beforeshifting into another mode of thinking. For instance, Carl’s 1 initial response was,“She would just start by finding one vector that was linearly independent, that couldnot be made as a combination of v and w, and once she found that vector, she wouldhave to find another one that was not a linear combination of the other three.” This wasthe only consistent use of Abstract thinking across the students in our study. Once theywere given specific numerical vectors, students tended to quickly shift into Computa-tional thinking.

We had anticipated that the notion of orthogonality would be most likely to leadstudents to think Geometrically, for example recommending that Michelle choosevectors orthogonal to those she already has. To our surprise, it was an intuitive ideaof probability, or “measure zero”, that led one student in this direction. For instance,when Bob was asked how many bases meet Michelle’s requirements, he said:

Bob: She would just pick a third vector that’s not in the same plane as the otherones. Which, if you’re talking about comparing infinities, we’re talking about alittle infinity for just that plane, versus the huge infinity that’s the rest of R3. So inR4 would be the same; there’s a little three-dimensional space, and that the rest ofR4 is huge, that you can choose from.

Here, Bob attacked the problem Geometrically, by visualizing the relative sizes ofR3 and R4. He had previously made similar comments about the odds that randomguessing would produce a suitable basis:

Bob: Well, it’s comparing two infinities, but like. If it’s R3, if they’re linearlydependent then they only form like a plane in R3. And there’s just one plane outof the infinitely many other planes that are in R3. Basically, it’s the odds of thethree points [sic; vectors is meant] all being in the same plane in R3, which R3 isjust like, it’s three-dimensional, and the odds that they’re all just in a two-dimensional slice of it is much lower. I mean, yeah, I guess you can’t come upwith odds, because they’re infinities, but maybe like 1% or something. [laughs]Probably less than that.

As discussed earlier, the occurrence of the word “plane” is a key indicator ofGeometric thinking; this was the first time it appeared in Bob’s interview.

Alan provides another example of Geometric thinking powering intuition. Whenasked to describe orthogonality, he uses the Abstract mode to provide a “technicaldefinition,” then describes his intuition in more Geometric terms with reference to“right angles:”

1 All student names are pseudonyms.


Alan: I usually think about it as being perpendicular, but I think the technicaldefinition is more that the dot product is zero. But I like thinking about it as thedirections in R3, just like your x and your y and your z pretty much. And then inRn, you just have n directions… they’re sort of at right angles. I don’t know howto think about that in Rn, but that’s what I think about; I just think about R3 anduse that as a basis for my thinking about Rn.

In contrast to the pilot study, where student thinking was largely balanced betweenall three modes of thinking, we noted a preponderance of Computational thinking in thepresent study; indeed, two students provided no evidence of Geometric thinking at all.As a crude indication, although the word “orthogonal” was used multiple times by eachstudent, the specifically Geometric word “perpendicular” was used by only two of theeight students, and the Geometric word “plane” was used by four of the eight. Incontrast, five of the eight students in the pilot study provided evidence of Geometricthinking in response to the Michelle question. This surprising (in view of the pilotstudy) preponderance of Computational thinking is what led us to expand our initial listof research questions to include a more focused question examining students’use of Computational thinking. Our data suggest that students using Computa-tional thinking can generate strategies, formulate justifications, reveal tacitassumptions, and provide several different ways to frame a solution process.We present vignettes of two students who exhibit some of these surprising andproductive ways of using Computational thinking. Through these vignettes, weillustrate some of the affordances and pitfalls of Computational thinking, whichwe summarize at the end of this section.

Bob: Strategy Generation and Computational Justification

To prompt students to justify their solutions, we included in our protocol the question,“Michelle is skeptical that your solution will always work. How would you convinceher?” We had anticipated that this question would prompt Abstract thinking, as this isthe language commonly used in theorems and proofs. However, we found that severalstudents were able to produce largely valid justifications using Computational thinkingalone. Additionally, several students’ Computational thinking inspired the creation ofstrategies for choosing vectors. The first vignette, featuring a student we call Bob,exemplifies both of these phenomena.

Bob used what we call the Missing Pivots strategy, which he came toformalize as follows: “Form an augmented 4x2 matrix with v and w and rowreduce to echelon form. From there, choose two new vectors so that thenonpivotal rows have a nonzero number, and all other numbers in the vectorare zero.” In particular, Bob used elementary basis vectors to supply themissing pivots.

Recall that, when a matrix is in reduced row echelon form, the leftmost nonzeroentry of any row must be 1, by definition. In the textbook used for this course, and thusin the classroom discourse, these entries are called “pivotal 1s” or “pivots” (some textscall them “leading 1s”). Accordingly, the rows or columns in which the pivotal 1s occurare called pivotal rows or columns. The pivotal columns of the reduced row echelonformMre of a matrixM correspond to linearly independent columns ofM. For example,


if columns 1, 3, and 4 of Mre are pivotal, then columns 1, 3, and 4 of M are linearlyindependent. In the context of the Michelle problem, the full set of column vectors ofMis independent if all columns are pivotal. Bob proposed augmenting the 4x2 matrixMre

with two additional columns so that each column (and row) in the resulting 4x4 matrixis pivotal.

Bob’s procedure was interesting to us in its own right for its sophisticationand novelty; this was not one of the methods we had anticipated studentswould use. However, Bob’s development of this procedure was also noteworthy.He first stated that “if you form a 4x4 matrix with [the vectors], it will rowreduce to the identity in R4. So, you need to choose vectors that are likely torow reduce to pivotal columns.” He suggested that “the odds are really good”that pure guess and check will accomplish this, and that including zeroes in theadditional vectors will be helpful: “she could include zeroes to make it easier,zeroes being a way to easily show that there’s no way you can write it as acombination of the other ones.” Specifically, he proposed adding two vectors ofthe form [0 0 # #] and [0 0 0 #], and successfully checked that this procedureworks in a specific numerical example. This solution strategy is reflective of arich conceptual understanding of the relationship between basis vectors andpivot columns, which Bob accessed here through reasoning Computationallyabout the row reduction process. Thus, Bob provides us with an example ofComputational reasoning as “integration of procedural knowledge withconceptual knowledge” (Baroody et al. 2007, p. 121).

In the course of formalizing this method for Michelle, it emerged that hewas tacitly assuming that the first two columns v and w of the matrix reduce tothe standard basis vectors e1 and e2. When the interviewer questioned thisassumption, suggesting a counterexample v=e1 and w=e3, Bob realized thathe needed to use the first two columns to ascertain which pivots are missingand must be supplied by the two additional vectors. Thus, the Missing Pivotsstrategy emerged as a refinement of simple guess and check methods motivatedby concrete counterexamples to an insufficiently general tacit assumption. Hisconstruction of the procedure could be described as reverse-engineering thecheck: by using Computational reasoning to think about the test his vectorsmust pass, he was able to engineer vectors that are guaranteed to pass it.Therefore, an affordance of Computational reasoning is the possibility of testingand refining methods and assumptions against well-chosen examples andcounterexamples.

Next, the interviewers asked him a question intended to elicit justification: “Michelleis skeptical that your method will work; how would you convince her?” Bob’sjustification was entirely Computational:

Bob: Well, by the definition of linear independence, their matrix has torow-reduce to the identity. All the columns will be pivotal. So, by usingthese facts about linear independence and pivotal columns, this procedureis a way to find two more columns that will be pivotal columns inde-pendent of the other ones already found. With the two vectors shealready has, she has two pivotal columns here, and they both representpivotal ones. It’s just a way to find the other two columns that won’t


form the same pivotal row as another one, so that they’ll all beindependent.

We note the presence of strongly Computational language such as “row-reduce” and“procedure.” Additionally, we argue that the notion of pivots on which this argumentrelies is so closely linked to the row reduction algorithm that it is a de facto sign ofComputational thinking.

This technique of justification by analysis of an algorithm is a particularly valuableway of constructing formal justifications. Many proofs in linear algebra proceed in asimilar fashion; for instance, the usual proof that more than n vectors in Rn cannot belinearly independent proceeds by forming a matrix whose columns are these vectors,row reducing, and arguing that the shape of the resulting echelon form implies that onevector is a linear combination of the others. The fact that students are capableof producing such justifications, as evidenced by Bob and others in our study,is perhaps an argument for teachers to highlight this method when discussingproof techniques.

It is worth noting that Bob’s justification falls slightly short of being a fully correctproof. In particular, say that the completed basis consists of the given vectors v and wand the additional vectors x and y. The vectors supplementing the row-reducedforms of v and w are technically the row-reduced forms of x and y, and shouldhave the inverse row operations applied to them to yield the original x and y.In Bob’s case, where the additional vectors are elementary basis vectors, theparticular inverse row operations that would be applied would leave the addi-tional vectors unchanged, but to be called a fully correct proof, his argumentshould have addressed this complication. Our data do not provide clear evi-dence of whether he was aware of this issue. In any event, the Computationaljustification Bob produced was correct as far as it went, and was most of theway to a complete proof.

Greg: Numerical Examples and Framing

The work of a student called Greg illustrates several more facets of students’productive Computational thinking. Here we first discuss Greg’s solution pro-cedure with minimal editorial comment, then offer two possible and relatedexplanations of his actions.

To orient the reader, we briefly discuss the mathematics underlying what we calledthe Unknown Columns strategy, illustrated by Greg in Fig. 1. This strategy proceeds asfollows: augment v and w together with vectors composed entirely of variables, androw reduce until the first two columns are e1 and e2. At this point, the third and fourthcolumns are partially row reduced. To continue the row reduction process from here,one would need to divide the third column by the diagonal entry; thus, for x to belinearly independent of v and w, it suffices that the diagonal entry be nonzero. This putsa constraint on the components of the original vector x; a similar constraint on thevalues of the original vector y emerges from continuing the row reductionprocess further.

As stated above, Greg used this Unknown Columns strategy: he augmented v and wwith vectors x and y composed entirely of variables (see Fig. 1), and performed row


operations on the resulting 4x4 matrix until the first two columns were reduced to e1and e2. He obtained the following matrix:

He then said, “These two [the diagonal entries] should be 1, and those [the off-diagonal entries] should be zeroes… because for these columns to be linearly indepen-dent, it would row-reduce to the identity matrix.” This reasoning gave him thefollowing four equations for the third column x, and similar equations for thefourth column y:

x1 ¼ 02x1−x2 ¼ 0x3−11x1 þ 4x2 ¼ 1x4−8x1 þ 2x2 ¼ 0

(These equations are overly restrictive, in that they are sufficient but not necessaryfor the third column to be linearly independent of the first two. In particular, the onlynecessary condition is that x3 – 11x1+4x2 is nonzero.)

The interviewers were interested to see how far he could push this line of reasoning,and asked him, “Can you give numerical vectors that satisfy these equations that you’vegot?” He obliged, and found that this system of equations uniquely determines thevector [0 0 1 0]; similarly, he concluded that his parallel set of equations for the fourthcolumn y uniquely determines the vector [0 0 0 1]. (This is the precise sense in whichthe equations Greg wrote are overly restrictive; they uniquely determine e3 and e4,which are valid solutions to this problem, but not the only ones.) This appeared toviolate his expectations, inasmuch as he asked, “Is there a way to take this back to thenon-row-reduced form?” He thus seemed to believe that [0 0 1 0] was the row-

Fig. 1 Greg’s unknown columns matrix, initial and row-reduced


reduced version of the original x vector, and was unsure how to recover theoriginal vector. He also said he was sure that this row-reduced version of x,denoted xr, is independent of vr and wr, but perhaps not independent of v andw themselves; further, he was sure that x (the original, “non-row-reduced”version) would be independent of v and w.

Later in the interview, the interviewers asked Greg to reflect on how hewould validate a proposed third vector, and gave him a numerical example, [1-1 0 0]. He augmented v and w with this vector and proceeded to row-reduceuntil his first two columns were e1 and e2. At this point, his third row was [00 | -15] (see Fig. 2 below). He seemed unsure how to interpret this result,saying, “I’m actually kinda confused about what this tells me. Um… Did Imake a mistake?”

Once the interviewers assured him that he had not made a computational error, hereasoned further:

Greg: Well, actually, if I continue row-reducing this, then I would get a 1 here [inplace of the -15 in the final column], and then that would make it unsolvable. …And then, that would mean that this can’t be a linear combination of these two. Sothen, it’s not in the span.

Int: And is that good or bad, for purposes of this problem?

Greg: That’s good. So then, this could be an additional vector for a basis.

Int: And just what about your answer says that?

Greg: If I continue row-reducing this, which is something that I alsoblanked out on, then I would get a 1 in the final column, and that statesthat 0 times whatever plus 0 times whatever, so on, equals 1, which is nottrue. So searching for solutions is what this is, so therefore there are nosolutions.

Next, the interviewers asked Greg to reflect on what this example would mean forhis original solution process (i.e., his Unknown Columns method). He realized that hehad imposed too many constraints on the variables:

Greg: I should have known that if this thing that ended up in the x3 spot [i.e., thediagonal entry], if this was equal to 1, then there’d be no solution. So this is whatI want to satisfy. … Really, that’s the only important thing, basically. This stuff

Fig. 2 Greg’s work on validating a proposed third vector


[i.e., the constraints on the non-diagonal entries] is not important. It’s just this[constraint on the diagonal entry].

Int: What do you mean it’s just that that’s important?

Greg: This is the only thing that would determine whether or not my new columnwould be in the span. … And actually it’s not just “equal to 1,” it’s “not equal tozero.” So it can be any number.

Greg then replaced his previous conditions with the new condition, depictedin Fig. 3.

It is visible in this Figure that Greg first wrote “=1” but then replaced this with “≠ 0”(thus replacing his original overly-restrictive condition with the correct necessarycondition discussed earlier). He then chose numbers for the xi’s that satisfied the givenconstraint, and seemed much more satisfied with this solution than with his previoussolution.

Now that we have recounted Greg’s story, we propose two related explanations forhis “a-ha moment” and subsequent correction of his strategy. First, the numericalexample may have shown Greg that he was not done with his row reduction process.When row-reducing his matrix, he stopped as soon as the first two columns were in theform he wanted, and seemed to believe that the matrix was then in reduced row echelonform and should thus be equal to the identity matrix; note, for instance, that he obtainshis initial equations by setting the third column of his matrix equal to [0 0 1 0], the thirdcolumn of the identity matrix. He did not seem to realize that there was more work heneeded to do to reach this point (i.e., dividing the third row by the diagonal entry tocreate a pivotal 1, etc.); this work may have been obscured by the algebraic complexityof his approach. In the numerical example, he obtained [1 3 -15 10] as his third columnat the same point in the row reduction process, which seems to have helped him realizethat the third column of his Unknown Columns matrix need not be equal to [0 0 1 0],allowing him to relax the constraints on the variables accordingly. It thus appears thatnumerical examples can illuminate what purely algebraic manipulations can obscure;however, both of these are examples of Computational thinking, since they bothinvolve reasoning about the row reduction process. It is thus evident that Computa-tional thinking can take many forms, and that one form can be used to help make senseof another. It is also interesting that both Greg’s error (not realizing that he was not donewith the row reduction process) and its resolution (by way of examining a specificnumerical example) came from his use of Computational reasoning.

Additionally, the algebraic complexity of Greg’s work may have hindered histhinking in another way. As shown by his desire to find the “non-row-reduced form”of his initial solution, he seemed to have lost track of the meaning of the variables heintroduced; by definition, they are the entries in the original matrix, not its reducedform, and he had already undone the row reduction process by solving the equations he

Fig. 3 Greg’s constraint


had accumulated in the third and fourth columns. Again, it appears that the numericalexample helped him realize what his variables had originally meant.

Second, once Greg realized that he was not done, his thinking shifted from solving asystem of equations (induced by setting his third column equal to [0 0 1 0]) to findingand satisfying constraints (induced by saying that the diagonal entry should be nonze-ro). This is an example of a shift in what we call a framing; that is, a student’s implicitmental expectation for how a computation should go. Greg shifted from thinking ofequations, where the expected action is to solve and the expected outcome is a specificsolution, to thinking of constraints, in this case inequalities, where the expected actionis to satisfy and the expected outcome is a range of possible solutions. Bothframings are useful at times, but for Greg, the framing as equation was lessappropriate for the situation and seemed to cause him difficulty. The framing asequation led him to the (apparently unsatisfactory) conclusion that adding e3and e4 should be the only solution to the problem; while this is a correctsolution, it is not the most general form. By imposing unnecessarily strongconstraints, he obtained an overly restricted answer; by shifting his framing andrelaxing the constraint, he obtained a much more general form.

Our use of the construct framing is broadly consistent with its prior use in educationresearch and other fields of social science. Hammer et al. (2005) define a frame (p. 98)as “a set of expectations an individual has about the situation in which she finds herselfthat affect what she notices and how she thinks to act.” In addition to frames for socialsituations, these authors mention epistemological frames that describe the methods anindividual expects to use to construct or justify knowledge. We suggest that studentshave expectations for the sequence of steps a computation will follow and the outcomesof at least some of these steps. This notion is linked to those of scripts, schemas,monitoring, and metacognition (see, e.g., Polya 1945; Davis 1984; Schoenfeld 1992;Asiala et al. 1996). The frame may include actions to be taken if the expectations arenot met, but a sufficient deviation from expectations may trigger a shift offraming instead.

The Unknown Columns solution method was attempted by three otherstudents, but not completed due to its algebraic complexity. One of thesethree students, Alan, exhibited behavior that we interpret as another shift inframing. In the course of row-reducing the matrix, he freely divided rows byalgebraic expressions that appeared in their pivotal locations, so as to obtainpivotal 1s there. He was not initially aware of the fact that this amounted toassuming the pivotal entries to be nonzero and thus the matrix to be invert-ible. When he arrived at the identity matrix as the row-reduced form, he wassurprised. His framing, or expectation, perhaps based on experience withsimilar problems when a basis for a subspace was to be found, was that therewould be free variables remaining at the end of the reduction, which could bechosen so as to obtain the desired basis. That is, he expected the constraintson the possible additional vectors to appear at the end of the computation, interms of the free variables, rather than along the way, in the requirements thatcertain entries be nonzero. His surprise caused him to reorient his framing andfocus his attention on the expressions he had divided by. He wasexperimenting with setting them equal to specific nonzero values when theinterviewer moved on due to time constraints.


Summary of Affordances and Pitfalls of Computational Thinking

We now move on from the close examination of these two vignettes to present somemore general results about how our students used Computational thinking. Rather thancomprehensively examine each student’s uses of Computational thinking, we herepresent as a more general summary a list of the affordances and pitfalls of Computa-tional thinking we identified in our students’ thinking, and provide brief examples ofeach. We noted the following affordances of Computational thinking:

1. Working out an example to provide a general orientation to an unfamiliar problem:before being presented with specific vectors for v and w, Alan invented his ownnumerical examples to see how the other two vectors might be related.

2. Searching for a known algorithm that applies to the situation, or evaluating theapplicability of a known algorithm: several students thought about using the Gram-Schmidt process, but eventually discarded this idea when they realized it required adifferent set of inputs than the situation afforded.

3. Recognizing when systems of equations have no solution, a unique solution, orinfinitely many solutions: Greg reasoned that his [0 0 | 15] row indicates that thereis no solution, so the vector is not in the span of v and w; another student, Hal, saidthat there are infinitely many bases that satisfy Michelle’s requirements becausethere is not just one solution to a certain equation.

4. Clarifying a more general approach with a numerical example: Greg was confusedby his Unknown Columns method until he worked through the validation of [1 -1 00] as a proposed third vector.

5. Making choices to simplify computations or reasoning: Bob proposed the strategicpositioning of zeroes to “make it easier” to show that the guessed vectors areindependent of the original vectors.

Naturally, Computational thinking was not a panacea; some students encountereddifficulties that emerged directly from their use of Computational thinking. Here aresome of the pitfalls of this way of thinking that we observed:

1. Thinking that coefficients in linear combinations must be integers: when checkingwhether or not a third vector was linearly independent of two others, Alan onlychecked integer combinations of the first two until challenged by the interviewer.Students’ default concept image of a “number” is often a (positive) integer, perhapsbased on experience with oversimplified textbook examples (the “natural numberbias;” see, e.g., Christou and Vosniadou 2012).

2. Dividing by variable expressions without imposing the constraint that they must benonzero: during the row reduction process in the Unknown Columns solutionmethod, Alan divided a row by the expression (x3–3x1) + 4(x2−2x1) whichappeared in the diagonal entry. He then continued row reducing until he obtainedthe identity matrix, and then became confused that he had no constraints on hisvariables, neglecting the fact that dividing by variable expressions would haveintroduced constraints earlier.

3. Becoming overwhelmed by sheer algebraic complexity, e.g., many variables: Carlattempted the Unknown Columns method (which introduces eight variables


representing the entries of these columns) but did not persist to a solution heregarded as satisfactory because dividing by the diagonal entry would lead tooverly complicated entries elsewhere.

4. Circular substitutions leading to vacuous statements: when attempting the orthog-onality approach described below, Carl made circular substitutions that led him tothe equation 0 = 0.

5. Narrow focus on algorithmic procedures: several students thought that Gram-Schmidt was the only way to produce orthogonal vectors.

6. Uncertainty about meaning of variables, leading to difficulty in interpreting results:as described above, Greg lost track of the meaning of his variables, leading him tobe confused about whether or not his result was row-reduced.

7. Failing to realize when an algorithm has (not) been run to its proper completion:Hal stopped row reducing before obtaining the identity matrix, leading him tomistakenly conclude that the two vectors he chose formed a linearly dependent setwith the first two.

8. Narrowly framing a computation in terms of solving an equation, leading to anexpectation that the solution will be unique. We interpreted Greg’s thinking as ashift away from this narrow framing. Other sorts of algorithms may be subject tothis pitfall as well; for example, the standard algorithm for computing a basis forthe null space (say) of a matrix may foster the belief that this is the only basis forthat space.

Many of these dangers will likely be familiar to any reader acquainted with students’thinking in linear algebra.

Responses to the Suggested Orthogonality Approach

Contrary to our expectations based on the pilot study, no student spontaneouslysuggested finding vectors orthogonal to both v and w. When the interviewer proposedthis approach (as having been mentioned by “another student”) most students agreedthat it would work and translated it into some set of equations but generally could notcarry it through to a complete solution. They also did not shift to Geometric thinkingfor guidance, but remained in the Computational or Abstract mode, and this likelycontributed to their difficulties.

One student wrote the equations stating that the new vectors should have zero dotproducts with the original pair, e.g., v x=0, but tried to manipulate them in that Abstractform as opposed to Computationally introducing the components of the vectors asvariables. Two were concerned that the initial vectors v and w were not orthogonal toeach other, perhaps thinking that orthogonality is a transitive property. Geometricthinking would clarify that the new vectors need to be orthogonal to the plane of vand w, the particular basis in this plane being irrelevant. An indicator of the absence ofGeometric thinking is that all students used the term orthogonal rather than perpen-dicular when reasoning about this proposed solution process. Students who formulatedthe equations in components could make further progress, sometimes obtaining apartial solution, but did not obtain a general solution or clearly understand how manysolutions there would be. Some did not write the complete set of orthogonalityrequirements, and one unnecessarily required the two new vectors to be orthogonal


to each other (making his system of equations nonlinear). In this approach to theproblem, purely Computational thinking in the absence of coordination with theGeometric mode was insufficient for most students, consistent with our originalhypothesis.

Discussion

Some authors, and some instructors, may view Computational thinking as mathemat-ically unsophisticated, a form of procedural knowledge in the narrow sense (Hiebertand Lefevre 1986). Our data show that this is not necessarily the case; the students inour study were able to use Computational thinking in a variety of sophisticated,productive, and reflective ways, including generating Computational justifications forclaims and making strategic choices to limit the complexity of their calculations. Ourwork thus joins the body of literature presenting an optimistic picture of studentreasoning in linear algebra (e.g., Possani et al. 2010; Wawro 2014; Wawro et al.2012), and provides evidence for the utility of Computational reasoning. We also hopeto have provided evidence for the analytic usefulness of including reasoning processeswithin the construct of Computational thinking.

The fact that students in our study largely remained in the Computational mode ofthinking throughout the interviews could be viewed as confirmation of Hillel’s andSierpinska’s observations that they have difficulty switching flexibly between modeseven when this would be helpful. However, the question remains, why did thesestudents use Computational thinking so much more than those in our pilot study(Wawro et al. 2011)? Both groups of students took the same course, with the sametextbook, and were interviewed at the end of the class. While our data provide nodefinitive answers, we can offer some conjectures. First, the class in the present studywas taught with a heavy emphasis on computations. Many of the central problems oflinear algebra were framed in terms of finding solutions to systems of equations; this isalso the approach taken by the textbook (Hubbard and Hubbard 2009). In contrast, thestudents in the pilot study were taught by a different instructor who takes a moreGeometric approach to the subject.

Second, the structure of the interview may have privileged Computational ways ofthinking. To minimize feelings of frustration and to smooth the way for students toproduce as much interesting mathematics as possible, we gave students the concretenumerical vectors fairly early on in the interview, which may have led students to moreComputational thinking than their natural inclination. We could well have waited untillater to provide the concrete vectors, and pushed harder for non-numeric solutions tothe problem.

It is also possible that our students have not yet developed the ability to useGeometric thinking in higher-dimensional contexts. Framing the Michelle problem inR4 may have precluded this mode of thinking. We could have explored this possibilityby asking them to re-situate the problem in R3.

We had hoped to investigate students’ use of multiple modes of thinking, and thenature of the coordination or interaction between them. What are the “boundaries”between the modes and how should thinking near these boundaries be characterized?What can prompt students to switch modes, and how do insights obtained in one mode


affect thinking in another? Do difficulties in coordinating representations betweendifferent modes outweigh the benefits obtained from flexibility? These questionscannot be answered with the data we have reported in this paper, and thus remain forfuture work. The interview protocol may need to be redesigned to provide moreopportunities for students to switch modes of thinking, or include questions thatactually prompt such a switch.

We may recommend a few pedagogical implications of our work. First, the successof students in constructing Computational justifications by analysis of algorithm (andthe frequent presentation of such proofs in linear algebra textbooks) may imply thatinstructors should foreground this technique in classes with a proof component.Textbooks normally justify each new algorithm with a proof that it does compute whatit claims to compute, and the structure of such proofs could be emphasized. Other usesof the technique, such as the proof by row-reduction that a linearly independent set inRn contains at most n vectors, could be highlighted. Second, our results suggest thatinstructors may wish to adopt a more positive orientation toward computational andprocedural thinking in general, notice when their students are using it productively, andhelp them grow more expert-like in the ways they use it. In particular, instructors maybe more explicit about the notions of framing and metacognition. Students could beencouraged to anticipate the output of an algorithm they employ, monitor its steps as itprogresses to check consistency with expectations, and consider an Abstract or Geo-metric viewpoint for making sense of unexpected results.

Finally, our data imply that Computational reasoning in linear algebra is an oppor-tunity for students to develop and utilize deep procedural knowledge (Baroody et al.2007). While not all Computational reasoning is particularly deep, and while oursample is composed of students in a high-achieving population and is therefore notrepresentative of the general population of linear algebra students, our results are anexistence proof. Our students’ use of procedural elements such as framing and reverse-engineering in the service of activities such as strategy generation, justification, andtroubleshooting is evidence that linear algebra students can, perhaps with appropriatescaffolding from their instructors, use Computational reasoning in ways that blendprocedural and conceptual knowledge. Our data contain many examples of what thesophisticated blending of procedural and conceptual knowledge might look like. Wehope that other researchers and practitioners will join us in recognizing and researchingproductive ways students might develop and use deep procedural knowledge.

Appendix 1: Interview Protocol

The task is posed in the relatively abstract form: Michelle would like to create a basisfor R4. She has already listed two vectors v and w that she would like to include in herbasis, and wants to add more vectors to her list until she obtains a basis. Whatinstructions would you give her on how to accomplish this? Students are providedwith this task in written form, and scratch paper to use.

After students have suggested a solution method, or if they are having difficultyproducing one: Suppose the specific vectors v and wMichelle has chosen are [1 2 3 4]and [0 -1 4 2]. Can you show how your procedure would work with these vectors?Would you like to revise your procedure after trying it with these vectors?


Questions to elicit formalization and justification: Michelle is only available viaemail; how would you write down your procedure to communicate it to her? SupposeMichelle would like to find another basis containing v and w; can she use your methodto find another? Michelle is skeptical that your method will work; how would youconvince her?

Questions for students who propose guess-and-check methods: Suppose you are avery unlucky guesser; is there a strategy for making better guesses? How does rowreduction tell you that the vectors form a basis? Suppose Michelle wants to guessadditional vectors one at a time; can she check the third vector before guessingthe next?

Suggesting an alternate method, potentially eliciting geometric thinking: Anotherstudent suggested that since orthogonal vectors are independent, Michelle should pickvectors orthogonal to those she already has. What do you think of this idea?

Additional questions: How many bases are there that meet Michelle’s requirements?Can you describe them? How many dimensions of freedom are there in choosing suchbases? Can Michelle begin with any two vectors v and w, or does she need to be carefulin choosing them?

Within the above framework, the interviewer was free to ask additional questionsexploring students’ thinking about the methods they employed. Although it was not hispurpose to help the students solve the problem, some of the questions caused studentsto reflect on difficulties they had encountered and were therefore of assistance.

Appendix 2: Solution Methods for the Michelle Problem

When we designed our study, we listed possible solution methods that we expectedstudents might use. These are named in boldface below, and are followed by theadditional unanticipated methods our students used. The numbers of students observedto employ each method is given. Note that some students used multiple approaches.

1. Guess and Check (5 students). Simply guess two vectors x and y to supplement vand w in forming a basis. Check that the set of four do form a basis, for example byrow reducing a 4x4 matrix having these vectors as columns. This method succeedswith probability one, that is, unless the guesses are very unlucky.

2. Abstract Proof (0 students). Since the span of v and w is two-dimensional, chooseany vector x not in this span. Then choose any vector y not in the three-dimensional span of v, w, x. This is a proof of existence of the desired basis, inthe Abstract language. To translate it into a practical algorithm, Computationalthinking is needed to specify a method for making the required choices. Forexample, the span of v and w can be characterized by row reducing a 4x2 matrixhaving those columns, to find a simple basis for this span. Then find by inspectiona vector that is not a linear combination of these basis vectors. Or, augment thismatrix with a third column of unknowns and row reduce to determine the condi-tions on the unknowns for this third column not to be a linear combination of thefirst two. Several students stated the Abstract definition of a basis, but did notappear to extend this into a proof by, for example, arguing that there are vectors inR4 outside the two-dimensional span of v and w.


3. Orthogonality (0 students). Find all vectors orthogonal to both v and w (Geomet-ric thinking), for example by computing the kernel of the 2x4 matrix having thoserows. Choose two independent vectors in this kernel to complete the desired basis.This idea was suggested to students during the interview, but none used itspontaneously. This was in contrast to the pilot study, in which many studentsproposed it as their initial response.

4. Modified Linear Combination (1 student). Take some linear combination of v andw, and alter one component of the resulting vector. This gives a vector x that is(almost certainly) independent of v and w. Repeat with some linear combination ofv, w, and x.

5. Unknown Columns (4 students). Create a 4x4 matrix whose columns are v, w, andtwo columns of unknowns representing the desired additional vectors x and y. Rowreduce to determine the conditions on the unknowns that make the four columnsindependent. Choose values satisfying these conditions.

6. Missing Pivots (2 students). Row reduce the 4x2 matrix having columnsv and w, noting which rows of the reduced matrix contain pivots. Thensupply two additional columns having pivots in the complementary rows(for example, two of the standard basis vectors would accomplish this).The reduced columns are then independent, and by reversing the se-quence of steps in the row reduction one obtains four independentcolumns v, w, x, y forming a basis. The choice of standard basis vectorsis particularly convenient because they will often not change under thereduction steps.

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